Yes, you can culture the bacteria from yogurt provided the label says it contains live, active cultures.
Here is a project from Sibuddies that tells you how to grow yogurt bacteria:https://www.sciencebuddies.org/science- ... #procedure
The two most common bacteria in yogurt are Lactobacillus sp. and Streptococcus thermophilus. Both are Gram positive. The first has the rod form of a bacillus while the second is a coccus.
If you want a Gram negative bacterium, I would use Escherichia coli (E coli). Its not a yogurt bacterium, however, and the only way you can be sure of getting E coli is to buy it from a biological supply company like Carolina Biologicals.
Be sure to handle and dispose of bacteria safely. The nutrient agar plates you use to grow the cultures can also grow some bad bugs and fungi, so wear dispo gloves and safety glasses and treat anything that comes in contact with bacterial cultures in a solution of 10% Clorox.
If you have more questions, please elt us know.